Again, in order to affect the TM's execution time, the "proof difference" must have a first step where your TM with CH takes a different step than the one without CH does. There can be no such first step. There is no room in the TM specification for that. It doesn't matter what you prove or don't prove, the TM "with CH" and the TM "without CH" must take the exact same steps.
You are messing up proof vs. truth. Your proof doesn't affect how long a TM runs for. Your axioms do not affect how long the TM runs for. Show me, in the state table of the TM, where the CH or any other axiom you use for analysis affects the truth table or the execution time of the TM. No amount of analysis can change how many steps a TM runs for. The numbers may be large, but they are finite and completely determined. The only thing it can affect is whether you can know the number, but the number exists regardless.
Even more concretely: Show me how your choice of axioms affects BB(4). Show me how it affects the number of steps, without redefining BB. Forget CH; take any axiom you like... for the analysis. Just don't affect the TM itself; that's a given for you. Take anything else you like and show me how it changes the BB(4) number. You can't. You can easily take axioms that will produce a wrong result; most notably we could just take as an axiom that BB(4) = -4. The resulting analysis will show that BB(4) is -4. It will also be wrong.
I am confident my explanation addresses OPs original question, which is whether two different (and consistent) formal systems can disagree about whether a given TM halts or not (and hence differ about what the value of BB(N) for some N), and the answer is yes they can. The source of that disagreement would come down to two different models of natural numbers where the theory that proves that a given TM halts only models non-standard natural numbers whereas the theory that proves that the given TM does not halt contains a standard model of the naturals.
What you're discussing is a completely different matter that while is interesting, does not address OP's issue and is something I already responded to by stating that one must be very careful to differentiate between what is actually true versus what can be proven by a formal system. You are blurring the lines between the two but that distinction is critical to understanding the point being made.
You are messing up proof vs. truth. Your proof doesn't affect how long a TM runs for. Your axioms do not affect how long the TM runs for. Show me, in the state table of the TM, where the CH or any other axiom you use for analysis affects the truth table or the execution time of the TM. No amount of analysis can change how many steps a TM runs for. The numbers may be large, but they are finite and completely determined. The only thing it can affect is whether you can know the number, but the number exists regardless.
Even more concretely: Show me how your choice of axioms affects BB(4). Show me how it affects the number of steps, without redefining BB. Forget CH; take any axiom you like... for the analysis. Just don't affect the TM itself; that's a given for you. Take anything else you like and show me how it changes the BB(4) number. You can't. You can easily take axioms that will produce a wrong result; most notably we could just take as an axiom that BB(4) = -4. The resulting analysis will show that BB(4) is -4. It will also be wrong.